Let $F$ be an algebraically closed field of characteristic zero, and let $A$ be an associative unitary $F$-algebra graded by a group of prime order. We prove that if $A$ is finite dimensional then the graded exponent of $A$ exists and is an integer.

A domain $\Omega $ is called a domain of injective holomorphy if there exists an injective holomorphic function $f\,:\,\Omega \,\to \,\mathbb{C}$ that is non-extendable. We give examples of domains that are domains of injective holomorphy and others that are not. In particular, every regular domain $(\overset{\multimap }{\mathop{\Omega }}\,\,=\,\Omega )$ is a domain of injective holomorphy, and every simply connected domain is a domain of injective holomorphy as well.

In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous neutral functional differential equations. An application to partial neutral differential equations is considered.

In this paper, we prove that a non–zero power series $F(z\text{)}\in \mathbb{C}\text{ }[[z]]$ satisfying

$$F({{z}^{d}})\,=\,F(z)\,+\,\frac{A(z)}{B(z)},$$

where $d\,\ge \,2,\,A(z),\,B(z)\,\in \,\mathbb{C}[z]$, with $A(z)\,\ne \,0$ and $\deg \,A(z),\,\deg \,B(z)\,<\,d$ is transcendental over $\mathbb{C}(z)$. Using this result and a theorem of Mahler’s, we extend results of Golomb and Schwarz on transcendental values of certain power series. In particular, we prove that for all $k\,\ge \,2$ the series ${{G}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1-{{z}^{{{k}^{n}}}})}^{-1}}$ is transcendental for all algebraic numbers $z$ with $\left| z \right|\,<\,1$. We give a similar result for ${{F}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1+{{z}^{{{k}^{n}}}})}^{-1}}$. These results were known to Mahler, though our proofs of the function transcendence are new and elementary; no linear algebra or differential calculus is used.

We prove the existence of an approximation function for holomorphic solutions of a system of real analytic equations. For this we use ultraproducts andWeierstrass systems introduced by J. Denef and L. Lipshitz. We also prove a version of the Płoski smoothing theorem in this case.

For the $n$-th order nonlinear differential equation, ${{y}^{(n)}}\,=\,f(x,\,y,\,y\prime ,\ldots ,\,{{y}^{(n-1)}})$, we consider uniqueness implies uniqueness and existence results for solutions satisfying certain $(k\,+\,j)$-point boundary conditions for $1\,\le \,j\,\le \,n\,-\,1$ and $1\,\le \,k\,\le \,n\,-\,j$. We define $(k;\,j)$-point unique solvability in analogy to $k$-point disconjugacy and we show that $(n\,-\,{{j}_{0}};\,{{j}_{0}})$-point unique solvability implies $(k;\,j)$-point unique solvability for $1\,\le \,j\,\le \,{{j}_{0}}$, and $1\,\le \,k\,\le \,n\,-\,j$. This result is analogous to $n$-point disconjugacy implies $k$-point disconjugacy for $2\,\le \,k\,\le \,n\,-\,1$.

We describe an ${{E}_{8}}$ correspondence for the multiplicative eta-products of weight at least 4.

The concept of ${{C}_{k}}$-spaces is introduced, situated at an intermediate stage between $H$-spaces and $T$-spaces. The ${{C}_{k}}$-space corresponds to the $k$-th Milnor–Stasheff filtration on spaces. It is proved that a space $X$ is a ${{C}_{k}}$-space if and only if the Gottlieb set $G(Z,\,X)\,=\,[Z,\,X]$ for any space $Z$ with cat $Z\,\le \,k$, which generalizes the fact that $X$ is a $T$-space if and only if $G(\sum B,\,X)\,=\,[\sum B,\,X]$ for any space $B$. Some results on the ${{C}_{k}}$-space are generalized to the $C_{k}^{f}$-space for a map $f\,:\,A\,\to \,X$. Projective spaces, lens spaces and spaces with a few cells are studied as examples of ${{C}_{k}}$-spaces, and non-${{C}_{k}}$-spaces.

In this paper we present a classification, up to equivariant isomorphism, of ${{C}^{*}}$-dynamical systems $(A,\,\mathbb{R},\,\alpha )$ arising as inductive limits of directed systems $\{({{A}_{n}},\,\mathbb{R},\,{{\alpha }_{n}}),\,{{\varphi }_{nm}}\}$, where each ${{A}_{n}}$ is a finite direct sum of matrix algebras over the continuous functions on the unit circle, and the ${{\alpha }_{n}}\text{s}$ are outer actions generated by rotation of the spectrum.

In this paper, we investigate a semilinear elliptic equation that involves multiple Hardy-type terms and critical Hardy–Sobolev exponents. By the Moser iteration method and analytic techniques, the asymptotic properties of its nontrivial solutions at the singular points are investigated.

We show that any Lipschitz projection-valued function $p$ on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions $q$ with Lipschitz constant close to that of $p$. This answers a question of Rieffel.

Following a similar result of Uspenskij on the unitary group of a separable Hilbert space, we show that, with respect to the lower (or Roelcke) uniform structure, the Polish group $G\,=\,\text{Aut(}\mu \text{)}$ of automorphisms of an atomless standard Borel probability space $(X,\,\mu )$ is precompact. We identify the corresponding compactification as the space of Markov operators on ${{L}_{2}}(\mu )$ and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on $G$, i.e., functions on $G$ arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that $G$ is totally minimal.

The Banach spaces $L(X,Y),K(X,Y),{{L}_{{{w}^{*}}}}({{X}^{*}},Y)$, and ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ are studied to determine when they contain the classical Banach spaces ${{c}_{o}}$ or ${{l}_{\infty }}$. The complementation of the Banach space $K(X,Y)$ in $L(X,Y)$ is discussed as well as what impact this complementation has on the embedding of ${{c}_{o}}$ or ${{l}_{\infty }}$ in $K(X,Y)$ or $L(X,Y)$. Results of Kalton, Feder, and Emmanuele concerning the complementation of $K(X,Y)$ in $L(X,Y)$ are generalized. Results concerning the complementation of the Banach space ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ in ${{L}_{{{w}^{*}}}}({{X}^{*}},Y)$ are also explored as well as how that complementation affects the embedding of ${{c}_{o}}$ or ${{l}_{\infty }}$ in ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ or ${{L}_{{{w}^{*}}}}({{X}^{*}},Y)$. The ${{l}_{p}}$ spaces for $1\,=\,p\,<\,\infty $ are studied to determine when the space of compact operators from one ${{l}_{p}}$ space to another contains ${{c}_{o}}$. The paper contains a new result which classifies these spaces of operators. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis.

In this article, we establish a new atomic decomposition for $f\,\in \,L_{w}^{2}\,\bigcap \,H_{w}^{p}$, where the decomposition converges in $L_{w}^{2}$-norm rather than in the distribution sense. As applications of this decomposition, assuming that $T$ is a linear operator bounded on $L_{w}^{2}$ and $0\,<\,p\,\le \,1$, we obtain (i) if $T$ is uniformly bounded in $L_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$; (ii) if $T$ is uniformly bounded in $H_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded on $H_{w}^{p}$; (iii) if $T$ is bounded on $H_{w}^{p}$, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$.

Let $\mathfrak{a}$ be an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules. Our main result asserts that if $R/\mathfrak{a}\,\le \,1$, then all generalized local cohomology modules $H_{\mathfrak{a}}^{i}(M,\,N)$ are $\mathfrak{a}$-cofinite.

We extend the notion of Zindler curve from the Euclidean plane to normed planes. A characterization of Zindler curves for general normed planes is given, and the relation between Zindler curves and curves of constant area-halving distances in such planes is discussed.

We provide a criterion for the central norm to be any value in the simple continued fraction expansion of $\sqrt{D}$ for any non-square integer $D\,>\,1$. We also provide a simple criterion for the solvability of the Pell equation ${{x}^{2}}\,-\,D{{y}^{2}}\,=\,-1$ in terms of congruence conditions modulo $D$.

For a finite module $M$ over a local, equicharacteristic ring $(R,\,m)$, we show that the well-known formula $\text{cd(m,}\,M)\,=\,\dim\,M$ becomes trivial if ones uses Matlis duals of local cohomology modules together with spectral sequences. We also prove a new ring-theoretic vanishing criterion for local cohomology modules.

In this paper we prove weighted norm inequalities with weights in the ${{A}_{p}}$ classes, for pseudodifferential operators with symbols in the class $S_{\rho ,\delta }^{n(\rho -1)}$ that fall outside the scope of Calderón–Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy–Littlewood type maximal functions. Our weighted norm inequalities also yield ${{L}^{p}}$ boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in $\text{OPS}_{\rho ,\delta }^{m}$.

Let $D$ be a connected bounded domain in ${{\mathbb{R}}^{n}}$. Let $0\,<\,{{\mu }_{1}}\,\le \,{{\mu }_{2}}\,\le \,\cdots \,\le \,{{\mu }_{k}}\,\le \,\cdots $ be the eigenvalues of the following Dirichlet problem:

$$\left\{ \begin{align} & {{\Delta }^{2}}u(x)\,+\,V(x)u(x)\,=\,\mu \rho (x)u(x),x\in \,D \\ & u{{|}_{\partial D}}\,=\,\frac{\partial u}{\partial n}\,{{|}_{\partial D}}\,=\,0, \\ \end{align} \right.$$

where $V(x)$ is a nonnegative potential, and $\rho (x)\,\in \,C(\overset{-}{\mathop{D}}\,)$ is positive. We prove the following inequalities:

$$\begin{align} & {{\mu }_{k+1}}\le \frac{1}{k}\sum\limits_{i=1}^{k}{\mu i+}{{[\frac{8(n+2)}{{{n}^{2}}}{{\left( \frac{\rho \max }{\rho \min } \right)}^{2}}]}^{1/2}}\times \frac{1}{k}{{\sum\limits_{i=1}^{k}{[{{\mu }_{i}}({{\mu }_{k+1}}-{{\mu }_{i}})]}}^{1/2}}, \\ & \frac{{{n}^{2}}{{k}^{2}}}{8(n+2)}\le {{\left( \frac{\rho \max }{\rho \min } \right)}^{2}}[\sum\limits_{i=1}^{k}{\frac{\mu _{i}^{1/2}}{{{\mu }_{k+1}}-{{\mu }_{i}}}}]\times \sum\limits_{i=1}^{k}{\mu _{i}^{1/2}}. \\ \end{align}$$